Local Smoothing Estimates near a Trapped Set with Infinitely Many Connected Components

TL;DR

This paper establishes local smoothing estimates for the Schrödinger equation on certain surfaces of revolution with infinitely many trapped geodesics, revealing how trapping complexity affects smoothing loss.

Contribution

It introduces a novel local smoothing estimate for surfaces with complex trapping, using semiclassical analysis and resolvent estimates, advancing understanding of wave behavior in trapping geometries.

Findings

Local smoothing estimate with loss depending on trapping complexity

Use of semiclassical propagation of singularities in trapping scenarios

Gluing resolvent estimates for infinitely many trapped geodesics

Abstract

We prove a local smoothing result for the Schr\"odinger equation on a class of surfaces of revolution which have infinitely many trapped geodesics. Our main result is a local smoothing estimate with loss (compared to \cite{ChMe-lsm}) depending on the accumulation rate of the critical points of the profile curve. The proof uses an h-dependent version of semiclassical propagation of singularities, and a result on gluing an h-dependent number of cutoff resolvent estimates.